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September  2019, 14(3): 589-615. doi: 10.3934/nhm.2019023

Optimal reinforcing networks for elastic membranes

Giovanni Alberti 1, , Giuseppe Buttazzo 1,, , Serena Guarino Lo Bianco 2, and  Édouard Oudet 3,

1. 

Dipartimento di Matematica, Università di Pisa, l.go B. Pontecorvo 5, 56127 Pisa, Italy
2. 

Dipartimento di Matematica e an, 80126 Napoli, Italy
3. 

Laboratoire Jean Kuntzmann, Université Grenoble Alpes, 38041 Grenoble, France

* Corresponding author

Received  August 2018 Revised  March 2019 Published  May 2019

In this paper we study the optimal reinforcement of an elastic membrane, fixed at its boundary, by means of a network (connected one-dimensional structure), that has to be found in a suitable admissible class. We show the existence of an optimal network, and observe that such network carries a multiplicity that in principle can be strictly larger than one. Some numerical simulations are shown to confirm this issue and to illustrate the complexity of the optimal network when the total length becomes large.

Keywords: Optimal networks, elastic membranes, reinforcement, relaxed solution, Golab's semicontinuity theorem.
Mathematics Subject Classification: 49J45, 49Q10, 35R35, 35J25, 49M05.
Citation: Giovanni Alberti, Giuseppe Buttazzo, Serena Guarino Lo Bianco, Édouard Oudet. Optimal reinforcing networks for elastic membranes. Networks & Heterogeneous Media, 2019, 14 (3) : 589-615. doi: 10.3934/nhm.2019023
References:
[1]

G. Alberti and M. Ottolini, On the structure of continua with finite length and Golab's semicontinuity theorem, Nonlinear Anal., 153 (2017), 35-55.  doi: 10.1016/j.na.2016.10.012.  Google Scholar

[2]

E. Acerbi, G. Buttazzo and D. Percivale, Thin inclusions in linear elasticity: A variational approach, J. Reine Angew. Math., 386 (1988), 99-115.  doi: 10.1515/crll.1988.386.99.  Google Scholar

[3]

M. Beckmann, A continuous model of transportation, Econometrica, 20 (1952), 643-660.  doi: 10.2307/1907646.  Google Scholar

[4]

G. Bouchitté, G. Buttazzo and P. Seppecher, Energies with respect to a measure and applications to low dimensional structures, Calc. Var. Partial Differential Equations, 5 (1996), 37-54.  doi: 10.1007/s005260050058.  Google Scholar

[5]

L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations, J. Math. Pures Appl., 93 (2010), 652-671.  doi: 10.1016/j.matpur.2010.03.010.  Google Scholar

[6]

G. Buttazzo, G. Carlier and S. Guarino Lo Bianco, Optimal regions for congested transport, ESAIM Math. Model. Numer. Anal., 49 (2015), 1607-1619.  doi: 10.1051/m2an/2015022.  Google Scholar

[7]

G. Buttazzo, É. Oudet and B. Velichkov, A free boundary problem arising in PDE optimization, Calc. Var. Partial Differential Equations, 54 (2015), 3829-3856.  doi: 10.1007/s00526-015-0923-1.  Google Scholar

[8]

G. Buttazzo, É. Oudet and E. Stepanov, Optimal transportation problems with free Dirichlet regions, in Variational Methods for Discontinuous Structures, Progr. Nonlinear Differential Equations Appl., 51, Birkhäuser, Basel, 2002, 41–65.  Google Scholar

[9]

G. Buttazzo and F. Santambrogio, Asymptotical compliance optimization for connected networks, Netw. Heterog. Media, 2 (2007), 761-777.  doi: 10.3934/nhm.2007.2.761.  Google Scholar

[10]

G. Buttazzo, F. Santambrogio and N. Varchon, Asymptotics of an optimal compliance-location problem, ESAIM Control Optim. Calc. Var., 12 (2006), 752-769.  doi: 10.1051/cocv:2006020.  Google Scholar

[11]

G. Buttazzo and N. Varchon, On the optimal reinforcement of an elastic membrane, Riv. Mat. Univ. Parma (Ser. 7), 4 (2005), 115-125.   Google Scholar

[12]

Y.-H. Dai and R. Fletcher, New algorithms for singly linearly constrained quadratic programs subject to lower and upper bounds, Math. Program. (Ser. A), 106 (2006), 403-421.  doi: 10.1007/s10107-005-0595-2.  Google Scholar

[13] K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986.   Google Scholar
[14]

S. Golab, Sur quelques points de la théorie de la longueur, Ann. Soc. Polon. Math., 7 (1929), 227-241.   Google Scholar

[15]

S. G. Johnson, The NLopt nonlinear-optimization package., Available from: http://ab-initio.mit.edu/nlopt. Google Scholar

[16]

S. J. N. Mosconi and P. Tilli, Γ-convergence for the irrigation problem, J. Convex Anal., 12 (2005), 145-158.   Google Scholar

[17]

E. Sánchez-Palencia, Nonhomogeneous Media and Vibration Theory, Lecture Notes in Physics, 127, Springer-Verlag, Berlin-New York, 1980.  Google Scholar

[18]

J. G. Wardrop, Some theoretical aspects of road traffic research, Proceedings of the Institution of Civil Engineers, 1 (1952), 325-362.  doi: 10.1680/ipeds.1952.11362.  Google Scholar

show all references

References:
[1]

G. Alberti and M. Ottolini, On the structure of continua with finite length and Golab's semicontinuity theorem, Nonlinear Anal., 153 (2017), 35-55.  doi: 10.1016/j.na.2016.10.012.  Google Scholar

[2]

E. Acerbi, G. Buttazzo and D. Percivale, Thin inclusions in linear elasticity: A variational approach, J. Reine Angew. Math., 386 (1988), 99-115.  doi: 10.1515/crll.1988.386.99.  Google Scholar

[3]

M. Beckmann, A continuous model of transportation, Econometrica, 20 (1952), 643-660.  doi: 10.2307/1907646.  Google Scholar

[4]

G. Bouchitté, G. Buttazzo and P. Seppecher, Energies with respect to a measure and applications to low dimensional structures, Calc. Var. Partial Differential Equations, 5 (1996), 37-54.  doi: 10.1007/s005260050058.  Google Scholar

[5]

L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations, J. Math. Pures Appl., 93 (2010), 652-671.  doi: 10.1016/j.matpur.2010.03.010.  Google Scholar

[6]

G. Buttazzo, G. Carlier and S. Guarino Lo Bianco, Optimal regions for congested transport, ESAIM Math. Model. Numer. Anal., 49 (2015), 1607-1619.  doi: 10.1051/m2an/2015022.  Google Scholar

[7]

G. Buttazzo, É. Oudet and B. Velichkov, A free boundary problem arising in PDE optimization, Calc. Var. Partial Differential Equations, 54 (2015), 3829-3856.  doi: 10.1007/s00526-015-0923-1.  Google Scholar

[8]

G. Buttazzo, É. Oudet and E. Stepanov, Optimal transportation problems with free Dirichlet regions, in Variational Methods for Discontinuous Structures, Progr. Nonlinear Differential Equations Appl., 51, Birkhäuser, Basel, 2002, 41–65.  Google Scholar

[9]

G. Buttazzo and F. Santambrogio, Asymptotical compliance optimization for connected networks, Netw. Heterog. Media, 2 (2007), 761-777.  doi: 10.3934/nhm.2007.2.761.  Google Scholar

[10]

G. Buttazzo, F. Santambrogio and N. Varchon, Asymptotics of an optimal compliance-location problem, ESAIM Control Optim. Calc. Var., 12 (2006), 752-769.  doi: 10.1051/cocv:2006020.  Google Scholar

[11]

G. Buttazzo and N. Varchon, On the optimal reinforcement of an elastic membrane, Riv. Mat. Univ. Parma (Ser. 7), 4 (2005), 115-125.   Google Scholar

[12]

Y.-H. Dai and R. Fletcher, New algorithms for singly linearly constrained quadratic programs subject to lower and upper bounds, Math. Program. (Ser. A), 106 (2006), 403-421.  doi: 10.1007/s10107-005-0595-2.  Google Scholar

[13] K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986.   Google Scholar
[14]

S. Golab, Sur quelques points de la théorie de la longueur, Ann. Soc. Polon. Math., 7 (1929), 227-241.   Google Scholar

[15]

S. G. Johnson, The NLopt nonlinear-optimization package., Available from: http://ab-initio.mit.edu/nlopt. Google Scholar

[16]

S. J. N. Mosconi and P. Tilli, Γ-convergence for the irrigation problem, J. Convex Anal., 12 (2005), 145-158.   Google Scholar

[17]

E. Sánchez-Palencia, Nonhomogeneous Media and Vibration Theory, Lecture Notes in Physics, 127, Springer-Verlag, Berlin-New York, 1980.  Google Scholar

[18]

J. G. Wardrop, Some theoretical aspects of road traffic research, Proceedings of the Institution of Civil Engineers, 1 (1952), 325-362.  doi: 10.1680/ipeds.1952.11362.  Google Scholar

Figure 1.  Approximation of globally optimal reinforcement structures for $ m = 0.5 $, $ L = 1, \, 2 $ and $ 3 $. The upper colorbar is related to the weights $ \theta $ which colors the optimal reinforcement set on the left, whereas the lower colorbar stands for the tangential gradient plotted on the connected set on the right picture
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Figure 3.  Approximation of globaly optimal reinforcement structures for $ m = 0.5 $, $ L = 4, \, 5 $ and $ 6 $. The upper colorbar is related to the weights $ \theta $ which colors the optimal reinforcement set on the left, whereas the lower colorbar stands for the tangential gradient plotted on the connected set on the right picture
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Figure 2.  Approximation of globaly optimal reinforcement strucutres for m = 0.5, L = 1.5, 2.5 and 5 for a source consisting of two dirac masses. The upper colorbar is related to the weights θ which colors the optimal reinforcement set on the left, whereas the lower colorbar stands for the tangential gradient plotted on the connected set on the right picture
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Table 1.  Reinforcement values computed on a fine mesh of $ 10^6 $ elements for classical and computed connected sets for $ m = 0.5 $
Length constraint Theoretical guesses Computed optimal networks
1 -0.179471 (radius) -0.178873
2 -0.165095 (diameter) -0.161944
3 -0.152676 (star) -0.149601
4 -0.141969 (cross) -0.138076
5 - -0.127661
6 - -0.117140
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Length constraint Theoretical guesses Computed optimal networks
1 -0.179471 (radius) -0.178873
2 -0.165095 (diameter) -0.161944
3 -0.152676 (star) -0.149601
4 -0.141969 (cross) -0.138076
5 - -0.127661
6 - -0.117140
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